If you move the tiles around, can you make squares with different coloured edges?

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Can you describe this route to infinity? Where will the arrows take you next?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Which set of numbers that add to 10 have the largest product?

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

Explore the effect of reflecting in two parallel mirror lines.

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

How many different symmetrical shapes can you make by shading triangles or squares?

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

Five children went into the sweet shop after school. There were choco bars, chews, mini eggs and lollypops, all costing under 50p. Suggest a way in which Nathan could spend all his money.

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

Different combinations of the weights available allow you to make different totals. Which totals can you make?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

How many solutions can you find to this sum? Each of the different letters stands for a different number.

Explore the effect of combining enlargements.

A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . .

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

Can all unit fractions be written as the sum of two unit fractions?

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

The clues for this Sudoku are the product of the numbers in adjacent squares.

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...

A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.